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Frequencies, Law-Governed Frequencies, and Ranges of Values John T. Roberts

Frequencies, Law-Governed Frequencies, and Ranges of Values John T. Roberts Department of Philosophy University of North Carolina at Chapel Hill johnroberts@unc.edu. What This Talk is About.

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Frequencies, Law-Governed Frequencies, and Ranges of Values John T. Roberts

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  1. Frequencies, Law-Governed Frequencies, and Ranges of Values John T. Roberts Department of Philosophy University of North Carolina at Chapel Hill johnroberts@unc.edu

  2. What This Talk is About • NomicFrequentism: An interpretation of objective probabilities I’ve been trying to develop and defend lately • The “Dynamical Probabilities Approach”: Strevens (2011), Abrams (2011) • These appear quite different, but… • …in fact, they’re a match made in heaven • Nomicfrequentism provides a solution to a problem posed for the dynamical probabilities approach, discussed e.g. by Rosenthal (2011) • Also, a merger with the dynamical probabilities approach helps nomic frequency with one of its problems – but that’s a topic for a different talk.

  3. The Dyamical Probabilities Approach • The probability of a given outcome in a macro-level process is equal to the fraction of the range of possible initial conditions that lead to that outcome • In Strevens’s terms: the probability of an outcome is its strike ratio WHEEL OF FORTUNE INITIAL CONDITION SPACE

  4. The Apparent Need for Input Probabilities • Half of the possible initial spin speeds lead to the outcome “Red.” • Therefore: The probability of the outcome “Red” is ½. • WHAT JUSTIFIES THIS INFERENCE? • ANSWER: AN ADDITIONAL PREMISE!

  5. The Apparent Need for Input Probabilities • Half of the possible initial spin speeds lead to the outcome “Red.” • The probability that a spin has one of the initial speeds that leads to a given outcome is equal to the size of the set of possible initial speeds that leads to that outcome. • Therefore: The probability of the outcome “Red” is ½ • BUT WHAT JUSTIFIES THIS ADDITIONAL PREMISE!

  6. The First Problem: • Suppose we seek a “metaphysics of probability”: • The dynamical probabilities approach: “merely describe[s] certain interesting cases of the transformation of probability distributions, but for this very reason cannot possibly be used for an analysis of probability. Given an initial-state space, and given a probability distribution on it, the outcome probabilities are fixed, but interpretational tasks of any sort are simply pushed back from the latter to the former probabilities… [T]he situation is still ‘probabilities in – probabilities out.’” (Rosenthal 2011 p.14)

  7. The Second Problem • The dynamical approach needs the input probabilities to satisfy certain assumptions • Namely: Macroperiodicity: the distribution must be approximately constant in any sufficiently small interval • THE TWO PROBLEMS: • 1. Interpret the input probability distribution • 2. Justify the assumption that the input probability is macroperiodic

  8. One way to address Problem 1: Interpret the input probabilities as objective chances, and adopt one of the existing philosophical theories of objective chances • Abrams and Strevens: No! • Instead let actual frequencies do the job of the input probabilities • To do the job, the initial-condition frequencies have to be approximately macroperiodic • That’s not enough: Also, their macroperiodicity must be counterfactually robust

  9. Abrams and Strevens: • If it had been the case that A, then it would most likely still have been the case that the initial-spin-speed distribution was still macroperiodic because: • By far most of the relevant nearby possible worlds where A is true are worlds where the distribution is macroperiodic.

  10. “I claim that a consequence of the way FFF mechanistic probability is defined in terms of bubbliness is that there is a natural sense in which there are more ways to preserve FFF mechanistic probability than to break it. …” (Abrams 2011, p. 22) “I have endorsed a natural extension of Lewis’s truth conditions to conditionals of the form If A had occurred at time t, then B would likely have occurred, true on a type 2 interpretation if B holds in nearly all the closest possible worlds in which A occurs at t.” (Strevens 2011, p. 27) “The closest worlds in which [a set of counterfactual coin-tosses] take place are therefore nearly all worlds with macroperiodic sets of initial speeds. The counterfactual conditional is true, then, as for similar reasons are the many other conditionals that diagnose the robustness of macroperiodicity.” (ibid p. 28)

  11. Abrams and Strevens: • If it had been the case that A, then it would most likely still have been the case that the initial-spin-speed distribution was still macroperiodic because: • By far most of the relevant nearby possible worlds where A is true are worlds where the distribution is macroperiodic.

  12. Abrams and Strevens assume: (3) The conditional “If it had been the case that A, then it would likely have been the case that C” is true whenever C is true at most of the closest possible worlds where A is true. Counterexample: There is a big machine at CERN, hooked up to a powerful particle accelerator. There is a red button on this machine, and if you push the button, then the machine will turn on, and it will flip a coin, in a biased manner, such that there is a 99% chance that it will land heads and a 1% chance that it will land tails. If the coin lands heads, then the machine will then initiate a process in the particle accelerator which will create very dangerous conditions that have a very high chance of causing all of the matter and energy in the universe to disappear. (4) If someone had pushed the button, then the universe would likely NOT have been annihilated.

  13. Abrams’s Other Argument: • “Thus breaking FFF mechanistic probability via transfers of inputs requires very special combinations of such transfers, which are reasonably considered ‘larger miracles’ (cf. Lewis 1979). This is the sense in which it’s easy to manipulate inputs in such a way that mechanistic probability is preserved, and difficult to manipulate inputs in such a way that it breaks.” (Abrams 2011, p. 23) • The basic idea: Counterfactually rearranging things to wreck macroperioidity would require changing lots of initial spin speeds; counterfactually rearranging things just a few spin speeds is a smaller miracle, and won’t wreck macroperiodicity. • But: Counterfactually changing one spin speed changes every later spin speed in an unpredictable way.

  14. From Actual Frequentismto NomicFrequentism ACTUAL FREQUENTISM: • The objective probability of getting heads when tossing a coin is ½ Is true iff: • Exactly one-half of all coin-tosses result in heads Problem: This deprives (5) of explanatory power and counterfactual robustness Solution: Identify (5) with (7) instead of (6): • It is nomologically necessary that half of all coin-tosses result in heads. This requires a law about frequencies – but why should that be a problem?

  15. From Actual Frequentismto NomicFrequentism Further problem: Is it nomologically necessary that there are a even number of coin tosses? Solution: Identify (5) with (8): (5) The objective probability of getting heads when tossing a coin is ½ (8) It is nomologically necessary that the frequency of heads among coin-tosses is in the interval [0.5 – f(N), 0.5 + f(N)] Here, N is the number of coin-tosses that ever occur, and f(N) is monotonic decreasing and approaches zero as N approaches infinity; for example, f(N) = 1/(2N)

  16. A Problem About Independence In addition to (5), (9) is true: (5) The objective probability of getting heads when tossing a coin is ½ (9) The probability of heads on a coin toss, conditional on the proposition that it is now raining in Paris, is ½ (or you could substitute for “it is now raining in Paris” any appropriate proposition – e.g. “the last 100 tosses resulted in Heads”) Obvious extension of what has been said so far: (9) is true just in case: (10) It is nomologically necessary that: The frequency of heads among coin tosses made while it is raining in Paris is within f(N) of ½ .

  17. Problem: That cannot work in general. Select a set of 10 coin tosses that all happen to land heads; describe the causal past of each one in minute microphysical detail, to distinguish it uniquely; then take the disjunction of the 10 descriptions, and call it X. The frequency of heads among X coin tosses is 1. So (11) is false: (11) It is nomologically necessary that: The frequency of heads among coin tosses made while condition X holds is within f(N) of ½ . But (11) should be true: X concerns the causal past of the coin-toss, so the result of the toss should be independent of whether X holds.

  18. The Solution: What it means for the probability of getting heads on a fair coin-toss to be ½, and for this to be independent of whether the last coin-toss resulted in head, and for it to be independent of whether it is raining in Paris, etc., is this: (12) It is nomologically necessary that the frequency of heads among coin-tosses is within f(N) of ½, and it is not nomologically necessary that the frequency of heads among coin-tosses made when the last coin-toss resulted in heads, or the frequency of heads among coin-tosses made when it is raining in Paris, etc., lies within any other interval. It might be true that the frequency of heads in coin-tosses made while it is raining in Paris is something other than ½ -- but if so, that’s nomologically contingent. Still though, it might have a nomological explanation… (“Every coin-toss made while it rains in Paris is of type Z; it’s a law that type-Z coin-tosses land heads with frequency ¾…”)

  19. (13) If Z is some condition that is not causally downstream from a given coin-toss, then for any value of p other than ½: there is no law that implies that the frequency of heads among fair coin-tosses made under condition Z is approximately equal to p (8) It is nomologically necessary that the frequency of heads among coin-tosses is in the interval [0.5 – f(N), 0.5 + f(N)] Suppose we are interested in the outcome of a certain coin toss made when it is raining in Paris (or right after a long run of heads, etc.); before observing the outcomes, we cannot have any inductive knowledge of the frequencies of heads among such coin tosses. So, the best information we have about THIS coin toss is that it is a coin toss, and (it is a law that) about half of them land heads. So we should make our predictions as if we knew that this coin toss had a chance of ½ of landing heads.

  20. I am assuming a principle of Direct Inference: When you know that: (i) x is an A; (ii) The fraction of As that are B is approximately r and it is also the case that: (iii) you don’t know anything else that bears evidentially on whether x is a B you are justified in believing to degree r that x is B. Now suppose x is a certain coin-toss that happens to follow a long run of tails results. Let A = coin-tosses, B = heads; r = ½. Then since we know (8) and (13), we know that (i) and (ii) are true, and also that (iii) is true – since nothing else we are in a position to know empirically is nomically linked to the frequency of heads among tosses like this one. So we are justified in having degree of belief ½ that the coin will land heads.

  21. NomicFrequentism = the view that there are laws about frequencies, and the objective probability statements found in the sciences shouyld be interpreted as laws about frequencies. Virtues: • Metaphysically parsimonious: Requires belief only in laws of nature (and you can take any view of lawhood you like) • Accounts for the counterfactual robustness and explanatory power of objective probabilities • Makes it easy to see how there could be autonomous higher-level probabilistic laws of the special sciences even if the fundamental laws are deterministic

  22. Returning to the problem confronting Abrams and Strevens: -The problem: How to find a way to let actual frequencies do the job of the input probabilities, without thereby making the dynamical probabilities too counterfactually fragile. -The solution: The relevant input frequencies are governed by laws of nature, and these laws require those frequencies to have whatever properties they need in order to support the dynamical properties.

  23. For example: Suppose the fundamental dynamics is classical. Let K be a kind of macro-level process, including spins of wheels of fortune The K-initial region is the region of the phase space of an N-particle system that is in the initial state of a K-type process. Let v be some phase-space volume; break up the K-initial region into v-sized cells; the actual frequencies will determine a frequency-distribution over these cells. (Its form will in general depend not only on v, but also on where we draw the boundaries between cells.)

  24. THE K-INITIAL REGION, DIVIDED INTO V-SIZED CELLS • PERHAPS: For some volume v and some way of breaking up into v-sized cells, the number stars in one cell is always approximately the same in adjacent cells • PERHAPS: For some volume of v, that’s true wherever you out the cell-boundaries • IN THAT CASE: Let’s say that the initial-condition distribution for K-type processes is macroperiodic relative to scale v. • IF NOMIC FREQUENTISM IS TRUE, then it might be a law that this is so.

  25. Psrt of the phase-space region of initial states of fortune-wheel spinnings by human croupiers (a subregion of the K-initial region):

  26. …divided into cells of volume u (the largest volume such that you can divide up the space into u-sized cells most of which are either all-red or all-black)

  27. Hypothesis: It is a law of nature that the frequency distribution over the K-initial region is macroperiodic relative to scale u • In that case: We should predict that about half of spins leads to result “Red.” This prediction is based only on laws, so it should be counterfactually robust.

  28. This prediction could go wrong: • But: Given our hypothesis, what we know about the laws tells us only that the distribution is macroperiodic relative to u, and by far the majority of possible distributions with that property lead to outcome-frequencies that match the strike-ratios.

  29. In Conclusion: If the hypothesis of a law about frequencies, to the effect that the frequency distribution over the K-initial region of phase space is macroperiodic relative to scale u, is true then: • We should predict the long-run frequencies to match the strike ratios, so we are justfiied in thinking of them as macro-probabilities; • We have not assumed an input-probability distribution of any kind (just an actual input-frequency distribution); • Nevertheless, the predicted outcome-frequencies are counterfactually robust, avoding the pitfalls of actual frequentism. This is just what Abrams and Strevens hoped to achieve; NomicFrequentism makes it possible.

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